The velocity of a particle moving along the $x$ -axis is $v(t)=3t+2$. At $t=0$, its position is $3$. What is the position of the particle, $s(t)$, at any time $t$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $s(t)=3$ (Choice B) B $s(t)=3t^2+2t+3$ (Choice C) C $s(t)=\dfrac32t^2+2t$ (Choice D) D $s(t)=\dfrac32t^2+2t+3$
Solution: We know that $s(t)= \int v(t) \,dt$. In this case, $s(t)= \int 3t+2 \,dt$ Let's find the indefinite integral: $\begin{aligned} \int 3t+2 \,dt&=\dfrac{3}{2}t^2+2t+C\\ \end{aligned}$ We know that $s(0)=3$. Let's use this information to solve for $C$. $\begin{aligned}s(t)&=\dfrac{3}{2}t^2+2t+C\\ \\ s(0)&=\dfrac{3}{2}(0)^2+2(0)+C\\ \\\\ 3&=C \end{aligned}$ The position of the particle $s(t)$ at any time $t$ is $s(t)=\dfrac32t^2+2t+3$.